Fourier multiplier norms of spherical functions on the generalized Lorentz groups

Our main result provides a closed expression for the completely bounded Fourier multiplier norm of the spherical functions on the generalized Lorentz groups SO0(1, n) (for n ≥ 2). As a corollary, we find that there is no uniform bound on the completely bounded Fourier multiplier norm of the spherical functions on the generalized Lorentz groups. We extend the latter result to the groups SU(1, n), Sp(1, n) (for n ≥ 2) and the exceptional group F4(−20), and as an application we obtain that each of the above mentioned groups has a completely bounded Fourier multiplier, which is not the coefficient of a uniformly bounded representation of the group on a Hilbert space. Introduction Let Y be a non-empty set. A function ψ : Y × Y → C is called a Schur multiplier if for every operator A = (ax,y)x,y∈Y ∈ B(l(Y )) the matrix (ψ(x, y)ax,y)x,y∈Y again represents an operator from B(l (Y )) (this operator is denoted by MψA). If ψ is a Schur multiplier it follows easily from the closed graph theorem that Mψ ∈ B(B(l(Y ))), and one referrers to ‖Mψ‖ as the Schur norm of ψ and denotes it by ‖ψ‖S. Let G be a locally compact group. In [Her74], Herz introduced a class of functions on G, which was later denoted the class of Herz–Schur multipliers on G. By the introduction to [BF84], a continuous function φ : G → C is a Herz–Schur multiplier if and only if the function (0.1) φ(x, y) = φ(yx) (x, y ∈ G) is a Schur multiplier, and the Herz–Schur norm of φ is given by ‖φ‖HS = ‖φ‖S. ∗Partially supported by the Ph.D.-school OP–ALG–TOP–GEO.